OVERHEATING DESPERATE FOR HELP

Hey my 1990 corolla is overheating to the point where the check engine light will come on and I will have to shut it off right away so it doesnt wreck it and there is no hot air blowing through but the a/c works fine.

The thermostat has been replaced twice in the past year and have just put a new radiator in it

Have been told it might be the heater core and if so how much might that be to replace or flush if needed

Flush the entire cooling system including the heater core, completely. A blocked heater core will not overheat the engine. The engine cools down when the thermostat opens allowing coolant to flow through the engine and through the radiator--where the heat is dissipated. If the coolant is not allowed to flow to the radiator when the engine warms up the engine will then over heat. The thermostat controls this flow. You have a sticking thermostat or a blocked radiator.
The heater core is a seperate coolant circuit from the engine coolant circuit but flush it anyway.
Also. Make sure the thermostat is not in backwards and the coolant is up to the right level.

The fact that there is no heat and the engine is overheating tells me that there is a problem with the coolant flowing from the engine to the radiator. The only thing that can restrict the flow is the water pump or a blockage in the cooling system. If you say that the cooling system was totaly flushed properly then the only thing it could be is the water pump. Just because it is new doesn't mean that it cannot have a defect. Also make sure the coolant level is full.